The amateur astronomical community has become
comfortable with a method of rating optical quality that is, quite
frankly, a very poor method indeed. It's time to drop the old Peak to
Valley (P-V) criterion and use a performance criterion common to the
professional, commercial optical world, the Strehl ratio. The Strehl
ratio has been in regular use for many years and will only be new to
many involved in amateur astronomy. The Strehl ratio is really quite
easy to relate to and will tell us a lot more about the performance of
an optic than a the P-V rating. Amateur astronomers would benefit
greatly by using a performance criterion that accurately and rationally
expresses a particularly important measure of optical performance,
rather than a statement of figure quality that is, despite its
convenient simplicity, ultimately unconnected to performance.

The Strehl ratio is simple to understand, as equally
simple as the current P-V standard, yet far more accurate and
comprehensive in intuitively describing the condition of an optical
wavefront. Although the mathematics involved in the fundamental makeup
of the Strehl ratio may be complex, they can be rather easily explained
in concept. In the end, the Strehl ratio may be far more appealing than
the old P-V standard when people become comfortable with it.

The amateur astronomical world is rapidly moving far
beyond the world that existed 10 or 15 years ago. Amateur astronomers
are demanding a new excellence in optical performance and those demands
extend themselves directly to the makers of optical components. They are
continuing to ask us for increasingly higher quality optics.
Unfortunately, these demands are being couched in terms that make the
production of such optics virtually impossible. It was one thing when
the standard was lambda/4 (1/4 wave) but quite another when the standard
is lambda/10. Human nature being what it is, claims become increasingly
exaggerated; particularly when the standard utilized has no intuitive
grounding in reality. If one person can produce lambda/10 optics then
another optician, far more skilled, can certainly produce lambda/20. And
if he can do that, the next wunderkind can produce lambda/40. And on and
on. But if you are wondering where this all ends, perhaps a more
important question is what it all means. And can anybody actually
achieve such numbers in terms of the common standard of peak to valley.

The significant advantage of the Strehl ratio is that
it is a measure of optical excellence in terms of theoretical
performance results, rather than an expression of the physical surface
or the shape of the wavefront. It is a complete statement, in terms of a
single number, that describes probably the most significant measure of
an optic's performance. In very simple terms, the Strehl ratio is an
expression of the amount of light contained within the Airy disk as a
percentage of the theoretical maximum that would be contained within the
disk with a perfect optical system. More correctly stated, it is the
measure of the fractional drop in the peak of the Airy disk as a
function of wavefront error. Strehl performance is usually expressed as
a range of numbers from 1 to 0, more rarely from 100% to 0%. A perfect
system is 1, a completely imperfect system is 0, and acceptable
standards occur somewhere in between. So, if someone says you have a
mirror that has a Strehl of .95 you understand that 95 percent of the
theoretical maximum amount of light is going where it should go. And 5
percent of that light is going into the surrounding rings (mostly and
most noticeably into the first order ring) and contributing to a
reduction in contrast. It's as simple as that. This is a concept that
can be easily understood by anyone once appropriate parameters have been
set and is a completely honest and accurate representation of optical
performance. But what are the parameters? What should we expect from a
really good optical system? Is .8 enough? Or should it be .9 or .95? And
how do these numbers relate, even approximately, to current peak to
valley standards? Perhaps the best way to answer this question is to
explore of the various commonly understood parameters for optical
excellence and see how they translate into a Strehl ratio.

The
Peak to Valley Criterion

To
demonstrate the weakness of the peak to valley method of measurement,
the following example is provided. The picture below is an exaggerated
cross-sectional representation of two hypothetical mirrors. Each mirror
is rated at lambda/10, peak to valley. Each mirror has a maximum and
minimum wavefront error, that when combined, equal a total amplitude
error of .10 wavelength of light. Yet, these mirrors are two totally
different optical surfaces and each will yield a vastly different
wave-form. At a casual glance, and without the benefit of sophisticated
analysis, which one would you rather have?

The answer is obvious. And so is the
inadequacy of the peak to valley methodology as a measure of optical
quality. It looks at only two points, the high and low, and ignores all
that lies between. Important issues such as roughness are totally
ignored while a very small high or low point are exaggerated totally out
of proportion to their significance. A mirror showing a peak to valley
value of lambda/2.5 (.4 wave) might in fact be an extremely good mirror
with the vast majority of its surface at better than lambda/10 and very
smooth. The reason that the peak to valley criterion is so popular is
due to a combination of factors that have transpired over the years.
When testing with a knife edge or a Ronchi screen, many errors that
would show up in interferometric testing are simply masked or subdued.
This is actually not a bad thing during the course of producing an optic
since knife edge and Ronchi testing adequately reveal correction error
and zones, which are the most serious problems encountered during
figuring if asymmetries such as astigmatism can be assumed to be under
control via good shop practices. For example, false astigmatism
generated by supporting an optic on its edge is completely unseen in the
knife edge and Ronchi tests and allows the optician to work without
confusion. Put under the more discerning light of an interferometer, the
same optic, held the same way, might show a large amount of astigmatism
which is not actually a permanent part of the optic and disappears when
the mirror is properly supported in the telescope. When estimates of
surface error are made with knife edge and Ronchi tests, largely by
measuring zones (which are relatively speaking fairly broad areas), a
peak to valley criterion is understandable and makes sense. In fact,
reducing the peak to valley error to negligible levels using this test
will usually assure a fine mirror. But under the scrutiny of the
interferometer, which measures very small areas over the entire surface
and not just a cross sectional slice through the middle, a more complete
analysis that integrates the results of many measures is necessary. The
essential point here is that the knife edge and Ronchi tests are
fundamentally different in their nature and application from that of
interferometery. Today's concept of measurement is based upon the
application of interferometry. Therefore, the new test demands a new
method of measurement appropriate to that test. Also, users of the knife
edge and Ronchi tests make a number of intuitive judgments above and
beyond mere peak to valley wavefront estimates. The optician also judges
the surface in terms of roughness and small zonal irregularities which
are usually reported as separate items in his analysis. A typical
comment might be, "1/10 wave, peak to valley, but rough and with a
small zone near the center." This is substantially different that a
blanket "1/10 wave" statement and injects auxiliary mitigating
statements that would not be part of an interferometric peak to valley
wavefront analysis. A typical interferometric analysis, aside from a
peak to valley wavefront analysis would also give indications of RMS
wavefront error as well as a Strehl ratio. And these might show a
significantly better optic.

The
RMS or Root Mean Squared Criterion

The RMS system of measurement is based
upon the principal of measuring a substantial amount of the optic's
surface at many points and then arriving at a single number that is a
statistical measure of the departure of the surface from the ideal form.
This technique puts into proper perspective any isolated areas which may
be in themselves highly deviated from the general surface. The name Root
Mean Squared is derived from the algorithm used to arrive at the final
statistic. This algorithm actually arrives at something known as
Standard Deviation, but because in optics the mean is always removed
from the error being rated, the RMS will always equal the standard
deviation. The process involves taking a set of readings over the
surface and then adjusting these readings equally, positively or
negatively, until the arithmetic mean of the readings are equal zero.
Likewise, tilts are removed when considering optics, since tilts only
represent a displacement of the image, not a degradation. Once this is
accomplished, all of the surface readings are then individually squared,
the squared readings are then averaged, and a square root is taken of
the average. For example, let's say that the following hypothetical ten
readings have been taken and represent deviations in wavefront of an
optic (see chart below). For illustrative purposes I have made the
second five readings the negative of the first five readings so that the
average of the readings will equal zero. It is immediately observed that
with the exception of two 1/5 wave readings, 80% of the surface appears
to hover around +/- 1/10th of a wave (1/5 wave, P-V), with 20% of the
surface at .4 wave. Yet, if a strict peak to valley criterion is used we
arrive at a reading for this mirror of .4 wave. Applying a RMS analysis
we arrive at the value of .125 or about 1/8 wave.

Data Points

1

2

3

4

5

6

7

8

9

10

Readings in
Waves

1/10
.1

1/12
.125

1/10
.1

1/9
.11

1/5
.2

-1/12
-.125

-1/10
-.1

-1/10
-.1

-1/5
-.2

-1/9
-.11

Readings Squared

.010

.016

.010

.012

.040

.016

.010

.010

.040

.012

Arithmetic Mean: 0
Sum of the Squares: .156
Average of the Sum of the Squares: .016
Square Root of the Average of the Sum of the Squares (RMS): .125

But does this mean that this is a 1/8
wave optic in any sense that one might ordinarily think? After all, most
people see "1/8 wave" and think, "peak to valley".
This is why the use of RMS as a sole measure of accuracy is dangerous.
While the amateur astronomical optical world has determined that 1/8
wave and 1/10 wave are desirable peak to valley standards for high
quality optics, no such commonly adopted standard has ever been
developed for RMS. Notwithstanding the fact that Rayleigh implied 0.074,
and Marechal and Marachal explicitly stated it, much confusion still
remains in the mind of the purchaser. The problem lies in the fact that
the RMS numbers are inherently more arbitrary and difficult to hang on
to than P-V standards, which at least have some basis in intuitive
reasoning. One can imagine a peak and a valley but not so easily a
standard deviation. Suggesting that 1/10 wave P-V might be equivalent to
about 1/35 wave RMS is likely not to stick in the mind of a telescope
buyer, but that actually is the approximate relationship when talking
about correction error, though no real relationship actually exists

The
Strehl Ratio

To reiterate, the Strehl ratio is a measure of optical
excellence in terms of theoretical performance results rather than an
expression of the physical surface or the shape of the wavefront, as in
P-V or RMS. The Strehl ratio is an expression of the amount of light
contained within the Airy disk as a percentage of the theoretical
maximum that would be contained within the disk with a perfect optical
system. It is the measure of the fractional drop in the peak of the Airy
disk as a function of wavefront error with a 1 Strehl being perfection
and anything less than 1 less than perfection. The Airy disk of an
unobstructed objective will contain a maximum of 83.8% of the energy
from that star entering the objective. The first order ring will contain
7.2% of the light, the second order ring 2.8%, and so on, diminishing
monotonically with each successive ring. Usually only the first ring is
clearly visible on a steady night. Basically, what it's all about is
getting as much of the light into the Airy disk as is possible.
Arguments arise about whether additional light in the first order ring
aids in the separating of double stars, but for all practical purposes
the more light in the Airy disk, the better off you are. Up to this
point, we have been discussing wavefront deviations related to large
figuring errors, but other factors make major contributions to optical
performance as well. One of these factors is surface roughness. With
knife edge and Ronchi testing surface roughness can only be judged
intuitively by a skilled worker making visual estimates according to his
experience. But with interferometry, roughness can actually be measured.
The ability to measure roughness in a quantitative way, along with all
other aberrations, is what makes interferometry and its associated
analysis tools so powerful in determining what constitutes the quality
of an optic. Analyzing an optic by measuring a large number of data
points (several hundred) results in an RMS measurement that contains the
data necessary to calculate the Strehl ratio and the impact of roughness
on the overall performance of an optic. The great value of the Strehl
ratio is that it expresses the powerful aspects of the RMS method in a
manner that is intuitive and memorable: 1 is perfection, .8 is okay. 9.
is good, .95 is extremely good. This kind of familiar and easy to
remember expression of quality is essential to creating unimpeachable
standards for optics.

The calculation of the Strehl ratio is not
particularly difficult once the RMS has been calculated. A simple
approximation formula is: Strehl Ratio = 1 - (2 pi * RMS) 2 and
works well for Strehls of high values. Applying this to the RMS of the
hypothetical surface calculated above, using the more exact and complex
formula of ((RMS * 2 * pi) 2 -/+ ) Inv Nat Log, we arrive at
Strehl ratio of .383. Not very good.

But why not very good? Just what is the Strehl standard?
What's good? Lets examine some known standards.

Quality
Bench Marks

Perhaps the best-known benchmark for optical quality is
the Rayleigh
limit, not to be confused with the Rayleigh Criterion, a
measure of the ability of the telescope to separate binary stars. The
Rayleigh limit basically asserts that if the wavefront reaching the eye
is distorted or deformed by spherical aberration more than 1/4
wavelength of yellow-green light, the image will be perceived as
degraded. Any wavefront reaching the eye having a de-formation of 1/4
wave or less will be perceived as essentially perfect. This standard has
been accepted as representing a minimum standard, not the highest
standard, but the minimum standard for high-quality optical performance.
Recent work suggests that perceptibly better performance is had by
having an optical system yielding a final wavefront through the eyepiece
of 1/8 wave. This more recent perception of the need for higher quality
optics has undoubtedly promoted recent passion for ultimate
optics. According to Suiter (Star Testing Astronomical Telescopes,
1994, page 9) the 1/4 wave Rayleigh tolerance as related to
spherical aberration causes a drop in the Strehl ratio to .8. Marechal's
work (Born and Wolf 1980, page 469) with respect to the RMS
statistic (the Marechal Criterion) has also shown that the Rayleigh
limit can be more generally expressed as an RMS wavefront of about 1/14
(.071), yielding a Strehl ratio of .82. This applies regardless of the
form of the error, as long as the RMS meets that level. But, of course,
more sophisticated and experienced observers tend to require a higher
level of quality, such that the total optical wavefront error should be
approximately half that of the old Rayleigh limit, or about 1/8 wave P-V
in the wavefront. Accepting Marechal's RMS of .071 as equivalent to
Rayleigh's .25 peak to valley, at least in the spirit that ATMs use the
Rayleigh limit, we arrived at the new RMS of .036 or a Strehl ratio .95
as a final 1/8 wave equivalent value. (The Rayleigh limit applies
strictly to spherical aberration. Other forms of aberration yield
different levels of P-V error to drop the Strehl to 0.80. This fact is,
as stated earlier, a critical flaw in the P-V rating.) If we desire a
final 1/8 wave accuracy for the system, and allowing a small error
accumulation due to the diagonal flat and eyepiece, not to mention the
eye itself, we see the derivation of the current standard of 1/10 wave
for a primary mirror (objective). From this, we can establish an
equivalent RMS of .028 (1/35.7). This translates into a Strehl ratio of
.97. A 1/9, P-V wave accuracy would yield an RMS of .032 or a Strehl
ratio of .96.

Reasonable
Strehl Standards

From this optician's viewpoint a reasonable standard for
telescopic objectives of high-quality would be those producing a Strehl
ratio from at least .95 upwards (1/8 wave, P-V if all correction error)
and preferably .96 (1/9 wave, P-V) with .97 (1/10 wave, P-V) as
representing the best that is reasonably attainable. The
purchaser of an optical system need only remember three pairs of
numbers: 1/8 wave = .95, 1/9 wave = .96, 1/10 wave = .97.
That's it. To make it even easier to remember, just remember that there
are three important P-V values (1/8, 1/9, 1/10) and to calculate
the equivalent Strehl one need only to subtract three from the
wavefront denominator to yield the second number of the Strehl value.
Example: 8 - 3 = 5, 1/8 wave = .95. This little shorthand device will
not work across the entire wavefront/Strehl ratio range, but it will
work for these three important numbers. A list of Strehls associated
with reasonably encountered P-V, wavefront, values are as
follows:

I sincerely hope that this article generates an interest
in utilizing a method of rating optics and optical systems that is not
only convenient but significantly superior to those already in use.

As a final note, the Strehl ratio derived
from an interferometric measurement of the RMS wavefront error
represents a ceiling on the attainable performance. In use, a mirror
will never attain this performance, mostly because of seeing, but also
because of collimation errors and errors from the other optical elements
in the path. Even the steadiest nights at the best sites do not allow a
system to achieve this performance, but it can come quite close.